\(2(8\mathrm{x}) + 4(7\mathrm{y}) = 12\)\(-2(8\mathrm{x}) + 4(7\mathrm{y}) = 12\)The solution to the given system of equations is \((\mathrm{x}, \math...

1. TRANSLATE the problem information

• Given system:
  • First equation: \(\mathrm{2(8x) + 4(7y) = 12}\)
  • Second equation: \(\mathrm{-2(8x) + 4(7y) = 12}\)
• Find: The value of \(\mathrm{8x + 7y}\)

2. INFER the most efficient approach

• Notice that the first equation has \(\mathrm{+2(8x)}\) and the second has \(\mathrm{-2(8x)}\)
• Key insight: Adding these equations will eliminate the 8x terms completely!
• This gives us a direct path to find 7y without needing individual x and y values

3. SIMPLIFY by adding the equations

• Add left sides: \(\mathrm{[2(8x) + 4(7y)] + [-2(8x) + 4(7y)]}\)
• Add right sides: \(\mathrm{12 + 12 = 24}\)
• Combine terms: \(\mathrm{2(8x) - 2(8x) + 4(7y) + 4(7y) = 24}\)
• Simplify: \(\mathrm{0 + 8(7y) = 24}\)
• Therefore: \(\mathrm{8(7y) = 24}\)

4. SIMPLIFY to solve for 7y

• Divide both sides by 8: \(\mathrm{7y = 24 ÷ 8 = 3}\)

5. SIMPLIFY to find 8x through substitution

• Substitute \(\mathrm{7y = 3}\) into the first equation: \(\mathrm{2(8x) + 4(3) = 12}\)
• Calculate: \(\mathrm{2(8x) + 12 = 12}\)
• Subtract 12: \(\mathrm{2(8x) = 0}\)
• Divide by 2: \(\mathrm{8x = 0}\)

6. SIMPLIFY to find the final answer

• Calculate \(\mathrm{8x + 7y = 0 + 3 = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the elimination opportunity and instead try to solve each equation individually for x and y. They might attempt to expand \(\mathrm{2(8x) = 16x}\) and \(\mathrm{4(7y) = 28y}\), then use substitution or other complex methods. This leads to unnecessary complexity and potential arithmetic errors, causing them to get confused and potentially guess or select an incorrect calculated result.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the elimination strategy but make sign errors when adding the equations, particularly with the \(\mathrm{-2(8x)}\) term. They might incorrectly get \(\mathrm{4(8x) + 8(7y) = 24}\) instead of \(\mathrm{0 + 8(7y) = 24}\). This arithmetic mistake propagates through their solution, leading them to calculate an incorrect value for the final expression.

The Bottom Line:

This problem rewards strategic thinking over computational complexity. The key breakthrough is recognizing that the structure of the equations allows for direct elimination, making the solution much simpler than it initially appears.